SOLVED.
I am self-studying from the Oskendal Stochastic Differential Equation and am trying to compute the density $B_{t}^{2}$ where $B_{t}$ is a one-dimensional Brownian motion.
I have $\mathbb{P}(B_{t}^{2} < x) = \mathbb{P}(B_{t} < \sqrt{x}) = \int_{- \infty}^{\sqrt{x}}p(y)dy = F(\sqrt{x})$
thus the density is given by $\frac{d}{dx} \mathbb{P}(B_{t}^{2} < x) = \Big(\frac{d}{d \sqrt{x}} \int_{- \infty}^{\sqrt{x}}p(y) dy \Big) \frac{d \sqrt{x}}{x}$
I don't know how to continue due to the $ \frac{d\sqrt{x}}{dx}$ outside the parenthesis.
I compute this to be $\frac{d}{d \sqrt{x}}F(x) \frac{d \sqrt{x}}{dx} = p(\sqrt{x}) \frac{d \sqrt{x}}{dx}$